Let´s suppose that we know all the possible causes for an outcome, for example:

**

# Bayes theorem

Posterior = ( Likelihood * Prior ) / Evidence

Here, P(movie|Sci-fi) is called Posterior, P(Sci-fi|Movie) is Likelihood, P(movie) is Prior, P(Sci-fi) is Evidence.

Prior: How probable was our hypothesis before observing the evidence? Posterior: How probable is our hypothesis given the observed evidence? Evidence: How probable is the new evidence under all possible hypotheses? Likelihood: How probable is the evidence give that our hypotheses is true?

The expected value is the mean of the posterior distribution.

$$ P ( A | B ) = \frac { P ( B | A ) P ( A ) } { P ( B ) } $$

We know that in pur favourite TV channel they show 40% of the times drama movies, 50% action and 10% horror movies, we´d like to know the probability that tonight there will be an horror movie P(A = Horror) = 0.1, this is our prior probability.

We check the Calendar and today is Hallowing´s night, in Hallowing the channels in general they show horror movies at least 75% of them P()

# Distributions

Discrete: They have only values based on a list of possible values, like the day of the month, number of persons… Continuous: They take arbitrarily exact values, for example prices, weights, distances… Mixed: A Combination of both categories.

# Mass and Density Functions

Discrete distributions have probability mass functions while continuous distributions have probability density functions.

## Probability Mass Function

The mass function for the Poisson distribution is:

$$ P ( Z = k ) = \frac { \lambda ^ { k } e ^ { - \lambda } } { k ! } , k = 0,1,2 , \ldots $$

## Probability Density Function

The density function for an exponential random variable:

$$ f _ { Z } ( z | \lambda ) = \lambda e ^ { - \lambda z } , \quad z \geq 0 $$

Th exponential variable can be continuous not only integers, still only positive numbers.

# Next

We’ll be using the following python packages: